L(s) = 1 | + (1.39 + 0.245i)2-s + (1.87 + 0.684i)4-s − 2.37i·5-s + (−1.53 − 2.15i)7-s + (2.44 + 1.41i)8-s + (0.582 − 3.30i)10-s + 0.335·11-s + 6.30·13-s + (−1.60 − 3.38i)14-s + (3.06 + 2.57i)16-s − 4.10·17-s − 2.19·19-s + (1.62 − 4.45i)20-s + (0.467 + 0.0825i)22-s − 6.23i·23-s + ⋯ |
L(s) = 1 | + (0.984 + 0.173i)2-s + (0.939 + 0.342i)4-s − 1.06i·5-s + (−0.579 − 0.815i)7-s + (0.866 + 0.500i)8-s + (0.184 − 1.04i)10-s + 0.101·11-s + 1.74·13-s + (−0.428 − 0.903i)14-s + (0.766 + 0.642i)16-s − 0.996·17-s − 0.502·19-s + (0.362 − 0.997i)20-s + (0.0997 + 0.0175i)22-s − 1.30i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.867 + 0.498i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.867 + 0.498i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.49277 - 0.665112i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.49277 - 0.665112i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.39 - 0.245i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (1.53 + 2.15i)T \) |
good | 5 | \( 1 + 2.37iT - 5T^{2} \) |
| 11 | \( 1 - 0.335T + 11T^{2} \) |
| 13 | \( 1 - 6.30T + 13T^{2} \) |
| 17 | \( 1 + 4.10T + 17T^{2} \) |
| 19 | \( 1 + 2.19T + 19T^{2} \) |
| 23 | \( 1 + 6.23iT - 23T^{2} \) |
| 29 | \( 1 - 4.38T + 29T^{2} \) |
| 31 | \( 1 - 8.10iT - 31T^{2} \) |
| 37 | \( 1 - 9.97iT - 37T^{2} \) |
| 41 | \( 1 + 7.35T + 41T^{2} \) |
| 43 | \( 1 - 0.892iT - 43T^{2} \) |
| 47 | \( 1 + 9.34T + 47T^{2} \) |
| 53 | \( 1 + 0.748T + 53T^{2} \) |
| 59 | \( 1 - 8.91iT - 59T^{2} \) |
| 61 | \( 1 - 7.33T + 61T^{2} \) |
| 67 | \( 1 + 5.56iT - 67T^{2} \) |
| 71 | \( 1 - 2.08iT - 71T^{2} \) |
| 73 | \( 1 + 5.81iT - 73T^{2} \) |
| 79 | \( 1 + 9.27T + 79T^{2} \) |
| 83 | \( 1 + 1.64iT - 83T^{2} \) |
| 89 | \( 1 - 11.3T + 89T^{2} \) |
| 97 | \( 1 - 2.81iT - 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.91315827523112451094001762305, −10.25861369031059975695834667618, −8.712673008470668876829896031748, −8.306531592767070980881324220257, −6.70094886638563226395185037622, −6.38306171299242944079905645788, −4.93180816850027648829711462257, −4.24577820114489198255280677094, −3.17469526938272077737151832136, −1.33894902010525887658994402325,
2.05194112810389253088259500612, 3.18407523402705892198138042690, 3.99303139391480432939627417828, 5.52729162809340894672523262504, 6.33377504595015639103763739462, 6.86528949895606004131395492043, 8.229544225487258851766857884493, 9.361978606965994323580274618335, 10.42715712016960339933634511275, 11.22359317649173653251704745999